Professor Ronald L. Carter ronc@uta.edu http://www.uta.edu/ronc/ Semiconductor Device Modeling and Characterization EE5342, Lecture 6-Spring 2004 Professor Ronald L. Carter ronc@uta.edu http://www.uta.edu/ronc/ L6 February 5

MidTerm and Project Tests MidTerm on Thursday 2/12 Cover sheet to be posted at http://www.uta.edu/ronc/5342/tests/ Project 1 draft assignment will be posted 2/13. Project report to be used in doing: Project 1 Test on Thursday 3/11 Cover sheet will be posted as above L6 February 5

Energy bands for p- and n-type s/c p-type n-type Ec Ev Ec EFi EFn qfn= kT ln(Nd/ni) EFi qfp= kT ln(ni/Na) EFp Ev L6 February 5

Making contact in a p-n junction Equate the EF in the p- and n-type materials far from the junction Eo(the free level), Ec, Efi and Ev must be continuous N.B.: qc = 4.05 eV (Si), and qf = qc + Ec - EF Eo qc (electron affinity) qf (work function) Ec Ef Efi qfF Ev L6 February 5

Band diagram for p+-n jctn* at Va = 0 Ec qVbi = q(fn - fp) qfp < 0 Efi Ec EfP EfN Ev Efi qfn > 0 *Na > Nd -> |fp| > fn Ev p-type for x<0 n-type for x>0 x -xpc -xp xn xnc L6 February 5

Band diagram for p+-n at Va=0 (cont.) A total band bending of qVbi = q(fn-fp) = kT ln(NdNa/ni2) is necessary to set EfP = EfN For -xp < x < 0, Efi - EfP < -qfp, = |qfp| so p < Na = po, (depleted of maj. carr.) For 0 < x < xn, EfN - Efi < qfn, so n < Nd = no, (depleted of maj. carr.) -xp < x < xn is the Depletion Region L6 February 5

Depletion Approximation Assume p << po = Na for -xp < x < 0, so r = q(Nd-Na+p-n) = -qNa, -xp < x < 0, and p = po = Na for -xpc < x < -xp, so r = q(Nd-Na+p-n) = 0, -xpc < x < -xp Assume n << no = Nd for 0 < x < xn, so r = q(Nd-Na+p-n) = qNd, 0 < x < xn, and n = no = Nd for xn < x < xnc, so r = q(Nd-Na+p-n) = 0, xn < x < xnc L6 February 5

Depletion approx. charge distribution +Qn’=qNdxn +qNd [Coul/cm2] -xp x -xpc xn xnc -qNa Charge neutrality => Qp’ + Qn’ = 0, => Naxp = Ndxn Qp’=-qNaxp [Coul/cm2] L6 February 5

Induced E-field in the D.R. The sheet dipole of charge, due to Qp’ and Qn’ induces an electric field which must satisfy the conditions Charge neutrality and Gauss’ Law* require that Ex = 0 for -xpc < x < -xp and Ex = 0 for -xn < x < xnc h 0 L6 February 5

Induced E-field in the D.R. Ex p-contact N-contact O - O + p-type CNR n-type chg neutral reg O - O + O - O + Exposed Acceptor Ions Depletion region (DR) Exposed Donor ions W x -xpc -xp xn xnc L6 February 5

Review of depletion approximation pp << ppo, -xp < x < 0 nn << nno, 0 < x < xn 0 > Ex > -2Vbi/W, in DR (-xp < x < xn) pp=ppo=Na & np=npo= ni2/Na, -xpc< x < -xp nn=nno=Nd & pn=pno= ni2/Nd, xn < x < xnc qVbi Ec EFp EFn EFi Ev x -xpc -xp xn xnc L6 February 5

Review of D. A. (cont.) Ex -xpc -xp xn xnc x -Emax L6 February 5

Depletion Approxi- mation (Summary) For the step junction defined by doping Na (p-type) for x < 0 and Nd, (n-type) for x > 0, the depletion width W = {2e(Vbi-Va)/qNeff}1/2, where Vbi = Vt ln{NaNd/ni2}, and Neff=NaNd/(Na+Nd). Since Naxp=Ndxn, xn = W/(1 + Nd/Na), and xp = W/(1 + Na/Nd). L6 February 5

One-sided p+n or n+p jctns If p+n, then Na >> Nd, and NaNd/(Na + Nd) = Neff --> Nd, and W --> xn, DR is all on lightly d. side If n+p, then Nd >> Na, and NaNd/(Na + Nd) = Neff --> Na, and W --> xp, DR is all on lightly d. side The net effect is that Neff --> N-, (- = lightly doped side) and W --> x- L6 February 5

Doping Profile If the net donor conc, N = N(x), then at x, the extra charge put into the DR when Va->Va+dVa is dQ’=-qN(x)dx The increase in field, dEx =-(qN/e)dx, by Gauss’ Law (at x, but also all DR). So dVa=-xddEx= (W/e) dQ’ Further, since qN(x)dx, for both xn and xn, we have the dC/dx as ... L6 February 5

Arbitrary doping profile (cont.) L6 February 5

Arbitrary doping profile (cont.) L6 February 5

Arbitrary doping profile (cont.) L6 February 5

Arbitrary doping profile (cont.) L6 February 5

n x xn Nd Debye length The DA assumes n changes from Nd to 0 discontinuously at xn, likewise, p changes from Na to 0 discontinuously at -xp. In the region of xn, Poisson’s eq is =*E = r/e --> d Ex/dx = q(Nd - n), and since Ex = -df/dx, we have -d2f/dx2 = q(Nd - n)/e to be solved L6 February 5

Debye length (cont) Since the level EFi is a reference for equil, we set f = Vt ln(n/ni) In the region of xn, n = ni exp(f/Vt), so d2f/dx2 = -q(Nd - ni ef/Vt), let f = fo + f’, where fo = Vt ln(Nd/ni) so Nd - ni ef/Vt = Nd[1 - ef/Vt-fo/Vt], for f - fo = f’ << fo, the DE becomes d2f’/dx2 = (q2Nd/ekT)f’, f’ << fo L6 February 5

Debye length (cont) So f’ = f’(xn) exp[+(x-xn)/LD]+con. and n = Nd ef’/Vt, x ~ xn, where LD is the “Debye length” L6 February 5

13% < d < 28% => DA is OK Debye length (cont) LD estimates the transition length of a step-junction DR (concentrations Na and Nd with Neff = NaNd/(Na +Nd)). Thus, For Va=0, & 1E13 < Na,Nd < 1E19 cm-3 13% < d < 28% => DA is OK L6 February 5

Ideal Junction Theory Assumptions Ex = 0 in the chg neutral reg. (CNR) MB statistics are applicable Neglect gen/rec in depl reg (DR) Low level injections apply so that dnp < ppo for -xpc < x < -xp, and dpn < nno for xn < x < xnc Steady State conditions L6 February 5

Apply the Continuity Eqn in CNR Ideal Junction Theory (cont.) Apply the Continuity Eqn in CNR L6 February 5

Ideal Junction Theory (cont.) L6 February 5

Ideal Junction Theory (cont.) L6 February 5

Law of the junction (cont.) L6 February 5

Excess minority carrier distr fctn L6 February 5

Excess minority carrier distr fctn L6 February 5

Forward Bias Energy Bands Ev Ec EFi xn xnc -xpc -xp q(Vbi-Va) EFP EFN qVa x Imref, EFn Imref, EFp L6 February 5

Carrier Injection ln(carrier conc) ln Na ln Nd ln ni ~Va/Vt ~Va/Vt ln ni2/Nd ln ni2/Na x -xpc -xp xnc xn L6 February 5

Minority carrier currents L6 February 5

Evaluating the diode current L6 February 5

Special cases for the diode current L6 February 5

Ideal diode equation Assumptions: Current dens, Jx = Js expd(Va/Vt) low-level injection Maxwell Boltzman statistics Depletion approximation Neglect gen/rec effects in DR Steady-state solution only Current dens, Jx = Js expd(Va/Vt) where expd(x) = [exp(x) -1] L6 February 5

Ideal diode equation (cont.) Js = Js,p + Js,n = hole curr + ele curr Js,p = qni2Dp coth(Wn/Lp)/(NdLp) = qni2Dp/(NdWn), Wn << Lp, “short” = qni2Dp/(NdLp), Wn >> Lp, “long” Js,n = qni2Dn coth(Wp/Ln)/(NaLn) = qni2Dn/(NaWp), Wp << Ln, “short” = qni2Dn/(NaLn), Wp >> Ln, “long” Js,n << Js,p when Na >> Nd L6 February 5

Diffnt’l, one-sided diode conductance Static (steady-state) diode I-V characteristic IQ Va VQ L6 February 5

Diffnt’l, one-sided diode cond. (cont.) L6 February 5

Charge distr in a (1- sided) short diode dpn Assume Nd << Na The sinh (see L12) excess minority carrier distribution becomes linear for Wn << Lp dpn(xn)=pn0expd(Va/Vt) Total chg = Q’p = Q’p = qdpn(xn)Wn/2 Wn = xnc- xn dpn(xn) Q’p x xn xnc L6 February 5

Charge distr in a 1- sided short diode dpn Assume Quasi-static charge distributions Q’p = Q’p = qdpn(xn)Wn/2 ddpn(xn) = (W/2)* {dpn(xn,Va+dV) - dpn(xn,Va)} dpn(xn,Va+dV) dpn(xn,Va) dQ’p Q’p x xn xnc L6 February 5

Cap. of a (1-sided) short diode (cont.) L6 February 5

General time- constant L6 February 5

General time- constant (cont.) L6 February 5

General time- constant (cont.) L6 February 5

Effect of non- zero E in the CNR This is usually not a factor in a short diode, but when E is finite -> resistor In a long diode, there is an additional ohmic resistance (usually called the parasitic diode series resistance, Rs) Rs = L/(nqmnA) for a p+n long diode. L=Wn-Lp (so the current is diode-like for Lp and the resistive otherwise). L6 February 5

Effect of carrier recombination in DR The S-R-H rate (tno = tpo = to) is L6 February 5

Effect of carrier rec. in DR (cont.) For low Va ~ 10 Vt In DR, n and p are still > ni The net recombination rate, U, is still finite so there is net carrier recomb. reduces the carriers available for the ideal diode current adds an additional current component L6 February 5

References * Semiconductor Physics and Devices, 2nd ed., by Neamen, Irwin, Boston, 1997. **Device Electronics for Integrated Circuits, 2nd ed., by Muller and Kamins, John Wiley, New York, 1986. ***Physics of Semiconductor Devices, Shur, Prentice-Hall, 1990. L6 February 5